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NCERT Solutions Ganita Prakash (Part 2) Chapter 4 Sierpinski Gasket — Figure it Out

Book page 72 Updated on2026-09-05

Q1.
Draw the initial few steps (at least till Step 2) of the shape sequence that leads to the Sierpinski Triangle.
Answer

Begin with a filled equilateral triangle. At every step, join the midpoints of each remaining triangle and remove the middle one.

Step 0 Step 1 Step 2 . . .
Step 1 removes one middle triangle; Step 2 removes a middle triangle from each of the 3 that survived.
Check it yourself: Step 2 should show 9 small filled triangles and 4 white holes — one large hole and three small ones.
Q2.
Find the number of holes, and the triangles that remain at each step of the shape sequence that leads to the Sierpinski Triangle.
Answer

Each remaining triangle gives 3 remaining triangles and 1 new hole at the next step.

Tn+1 = 3 Tn, with T0 = 1 ⇒ Tn = 3n
Hn+1 = Hn + Tn, with H0 = 0
⇒ Hn = 1 + 3 + 3² + … + 3n−1

Closing up the sum the same way as for the carpet:

2Hn = (3 − 1)(1 + 3 + … + 3n−1) = 3n − 1
Hn = (3n − 1)⁄2
Step n01234n
Triangles remaining13927813n
Holes0141340(3n − 1)⁄2
Why the two rules are different in shape: triangles are replaced at every step, so their count is multiplied — that gives a power. Holes are never removed, so their count is a running total — that gives a sum. Recognising which of the two is happening is the whole skill here.
Q3.
Find the area of the region remaining at the nth step in each of the shape sequences that lead to the Sierpinski fractals. Take the area of the starting square/triangle to be 1 sq. unit.
Answer

Carpet: (8⁄9)n sq. units. Gasket: (3⁄4)n sq. units.

Sierpinski Carpet. Each square is cut into 9 equal parts, so a square of Step n has area (1⁄9)n, and there are 8n of them.

Arean = 8n × (1⁄9)n = (8⁄9)n
Area1 = 8⁄9, Area2 = 64⁄81, Area3 = 512⁄729 ≈ 0.70

Sierpinski Gasket. Each triangle is cut into 4 identical parts, so a triangle of Step n has area (1⁄4)n, and there are 3n of them.

Arean = 3n × (1⁄4)n = (3⁄4)n
Area1 = 3⁄4, Area2 = 9⁄16, Area3 = 27⁄64 ≈ 0.42

A second route to the same answer: at each step a fixed fraction of whatever is left is thrown away — 1⁄9 of it for the carpet, 1⁄4 for the gasket. So the area is multiplied by 8⁄9 (or 3⁄4) every step.

Why it happens: both 8⁄9 and 3⁄4 are less than 1, so the area shrinks at every step and keeps shrinking. (3⁄4)10 ≈ 0.056 and (3⁄4)50 is under a millionth. The true fractal — the shape you approach after infinitely many steps — has zero area, even though something is still there at every point of it. Notice also that the carpet shrinks more slowly than the gasket, because 8⁄9 is bigger than 3⁄4.
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